(The different games in the series are basically upgraded, tweaked, and re-released versions of the previous game, not true sequels.)

Street Fighter Alpha (series)

Versions: the whole series (original, 2, 3)
Systems: Arcade.  Never played it on home systems.

A solid outgrowth from Street Fighter II. It added some new game mechanics that spruced up gameplay a bit, yet it managed to stay away from the combo-itis of Killer Instinct and Mortal Kombat 3. Charlie is awesome, although the fact that he's doomed to die if you beat the game is a bit discouraging -- his "joke" ending in SFA3 is fun, though.

Mortal Kombat II

Systems: Arcade.  Period.  All versions on home systems have been middling to terrible.

The best of the Mortal Kombat series by most measures. Not as timeless as the Street Fighter series, but it still has a warm fuzzy place in my heart.

Samurai Showdown (series)

Versions: Samurai Showdown 1 and 2.
System: Arcade (Neo-Geo).

One of my favorites from the Neo-Geo. Amusing premise, fun gameplay, and has some of the most memorable characters to be found in a fighting game (e.g. Galford, Nakoruru, Amakusa). Charlotte was my character of choice in the first game, but playing as Galford in the second one was too much fun to pass up. Galford's attacks using his pet dog Poppy are both effective and hilarious. "Secret Ninja Diving Falcon Death Teleport, Poppy!"

The later games in the series, though, just weren't as enjoyable. The third and fourth installments were both disappointing to me, and I didn't hold my breath for the fifth (a wise choice once it was released).

Tekken (series)

Versions: pretty much any of them.
System: Arcade.

The classic 3D arcade beat-em-up. It's best to just ignore the storyline, but the gameplay itself is fun. Law is always fun for me to play as (fast and nimble, very combo-friendly), as is Paul (slow but hard-hitting). Other characters can also be fun, depending on which incarnation I'm playing. Incidentally, Law and Paul make an awesome team in Tekken Tag, because they can cover each other's weaknesses.

Soul Calibur (series)

Versions: Soul Edge, Soul Calibur II
Systems: Arcade for Soul Edge.  Only played Soul Calibur II on GameCube.

Another classic 3D arcade beat-em-up, but one with a little more finesse than the Tekken series. The storyline is confusing, but unlike Tekken it clearly has a storyline. Li Long, Seung Mina, and Voldo were fun in Edge, while Raphael was my favorite from Calibur 2. I've never played Calibur 1 or 3.

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Shooters

Rez

System: PS2.

On the surface, Rez is your basic shooter-on-rails: your "ship" travels through the game world on a predetermined path, and your only control over the game world is aiming your turret to shoot at enemies. However, Rez breaks that mold very quickly. Rez is set in a bizarre but beautiful Virtual Reality, and the enemies you destroy are visual representations of malevolent "firewalls" created by a benevolent but misguided newborn AI, whom you must now rescue from her own creations (which she willed into existence in a moment of panic).

The big distinguishing trait of this game is that everything, and I mean everything, is set in time with the music: the game was trying to capture some of the experience of synaesthesia. The appearance of new enemies and obstacles, the timing of your own attacks, the rhythmic pulsing of the stage itself, and the constant rumble of the controller is all subordinate to the techno-trance beats that permeate the game. When you combine all that with the final stage that attempts to re-create a secular version of religious ecstasy, it's quite an experience.

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Adventure/Exploration

Metroid Prime

System: GameCube.

I love this game. While its predecessor, Super Metroid, is superior in some ways, I consider Metroid Prime to be the better game overall, albeit by a slim margin. Like everyone else, I was apprehensive when I heard that they were working on a 3D Metroid game, but the end result was far better than I'd expected.

In terms of gameplay, Prime starts off by borrowing a lot from the First Person Shooter and 3D Platformer genres. However, the lock-on system breaks away from the FPS emphasis on precision aiming, and the collision physics makes jumping from platform to platform feel more like Olympic gymnastics and less like a kindergartner falling off a jungle gym. The end result is that the player is allowed to worry less about the enemies and platforms, and more about solving puzzles and exploring what's out there.

Prime follows the modern trend of longer, more involved games. In contrast with Super Metroid, which takes perhaps 2 to 4 hours to beat, my first play through Metroid Prime took a bit over 20 hours. Also unlike Super Metroid, Prime rewards you more for being thorough than fast. Personally, I think this works better for the "exploration" angle.

When it comes to plot and character, Prime is very much of the mind that "Less is More", an ideal it shares with the older Metroid games. While the game will give you gentle nudges if you don't know where to go next, the plot feels more relaxed than forced, and you're always free to wander off to the Hall of the Elders for some quiet reflection, or to observe the wildlife in Phendrana Drifts. Likewise, none of the characterization is up-front; there is no dialogue in the game, and all character development comes from the bits of arcana that you find scattered across Tallon IV, the planet you're exploring. The result is that the world itself almost feels like the most important character after Samus, the heroine you play as. What character development occurs in Samus is largely implicit and mostly a creation of the player's imagination, flowing as it does from interpreting the history and present condition of Tallon IV through the lens of her backstory.

Metroid Prime 2: Echoes, on the other hand, is not nearly as good as the first in the series.  The enemies are palette swaps of Metroid Prime enemies (despite taking place on another planet, Aether); the map layout is uninspired, with recurring map patterns shared between entirely unrelated zones (e.g. a swamp and a fortress); the puzzles are uninspired, with some clearly being the result of bored level designers; and the difficulty of "Normal" mode is about as hard as "Hard" mode from the first game, for no good reason.  To add insult to injury, the plot of the second game pulls a minor Matrix by making the plot of the first game seem dumber in retrospect.  Basically, unless you're a completionist or you're looking for a "master quest" version of the first game, Echoes is not a high recommendation.

I haven't yet played Metroid Prime 3: Corruption, but the plot sounds even worse than Echoes while the gameplay sounds rather fun.  I intend to buy a Wii plus Metroid Prime: Trilogy at some point, as I've finally made peace with the plot direction.

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Real-Time Strategy

StarCraft

System: PC.

It's still one of the best RTS games out there, even though it's over a decade old.  A band of humans, ejected from Earth-controlled space as criminals and/or political malcontents, find their own interplanetary squabbles interrupted by the sudden appearance of two alien races: the violent, insectoid Zerg and the enigmatic, humanoid Protoss.  The game is famous for its multiplayer, which is still being competitively played to this day (especially in South Korea).

Warcraft 3

System: PC.

I never played the first two Warcraft games, but I enjoyed this one immensely. The gameplay isn't quite up to the elegance of StarCraft (four races makes it harder to balance), but the storyline is very engrossing.

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Role Playing Games

StarTropics

System: NES.

A very quirky game. Like the Legend of Zelda series, it's sort of a hybrid between an adventure game and an RPG. However, the gameplay itself is deliberately humorous and very unconventional -- your primary weapon is an ancient tribal yo-yo, and you solve puzzles by jumping on special floor tiles over expanses of lethal water. Just as you think the game can't get any weirder, a sudden plot twist comes out of nowhere for chapters 7 and 8.

The game is classic Nintendo at its best. For instance, one of the materials that comes with the game, attached to the user's manual, is a yellow letter from Dr. J, the protagonist's recently missing archaeologist uncle. By all appearances, the letter is nearly useless, except as an introduction to the plot. However, at a memorable point in Chapter 4, Dr. J's assistant tells you that your uncle, knowing that he was likely to be kidnapped, is carrying a transmitter and sent you the radio frequency. When you dip the actual letter in water, as instructed, it reveals the hidden message, allowing you to continue the game.  (Hint for those who lost the letter or are playing in emulation: it's 747 MHz.)

Paper Mario: The Thousand Year Door

System: GameCube.

A shockingly fun game. It blends some classical RPG gameplay elements (e.g. a world map and a separate battle arena) with some traditional Mario elements (e.g. jumping on enemies). But not in the way you'd expect, oh no. The actual combat is turn-based and requires more strategizing than dexterity; instead, you actually jump on or whack at enemies on the world map, of all places, which can give you a freebie "surprise" round at the start of the battle. Add the "badge" system to that, and you get an RPG with a surprisingly intelligent combat system.

Besides the actual combat mechanics, the strong point of the game is the humor. The jokes are very strongly postmodern, with lots of self-referential jokes (e.g. an NPC in one town is playing the same Paper Mario game on his own GameCube at home, giving you tantalizing but spoiler-free hints about the ending) and meta-breaking humor (e.g. crows that, when they think no one else is around, discuss geopolitical events and environmental issues from the player's Real World). Goombella is, by far, my favorite character, as she's constantly making meta-referential jokes that poke at the game itself. Whenever the game doesn't really make much sense, such as the party walking around on the Moon with no space suits, Goombella is the one who comments on how, by all rights, it ought to be impossible.

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MMORPGs

World of Warcraft

System: PC.

It's the only mainstream MMORPG that I'll give the time of day to. Not quite Nirvana, but it's fun nonetheless.  I have a Level 80 Tauren Druid (Bandanu on Deathwing US) and a Level 80 Tauren Hunter (Monani on Deathwing US).